On the Optimality of Lattices for the Coppersmith Technique

Abstract

We investigate the Coppersmith technique [7] for finding solutions of a univariate modular equation within a range given by range parameter U. This paper provides a way to analyze a general type of limitation of the lattice construction. Our analysis bounds the possible range of U from above that is asymptotically equal to the bound given by the original result of Coppersmith. To show our result, we establish a framework for the technique by following the reformulation of Howgrave-Graham [14], and derive a condition for the technique to work. We then provide a way to analyze a bound of U for achieving the condition. Technically, we show that (i) the original result of Coppersmith achieves an optimal bound for U when constructing a lattice in a standard way. We then show evidence supporting that (ii) a non-standard lattice construction is generally difficult. We also report on computer experiments demonstrating the tightness of our analysis. Some of the detailed arguments are omitted due to the space limit; see the full-version [1].